Related papers: Asymptotic energy of graphs

We introduce the matrix sums that represent a discrete analog of the matrix models with quartic potential. The probability space is given by the set of all simple n-vertex graphs with the Gibbs weight determined by the graph Laplacian. We…

The asymptotic form of the energy density for a gas of particles surrounding a sphere of mass $M$ and radius $R$ is studied using Einstein's equations. It is shown that if the pressure of the gas $p$ varies linearly with the energy density…

Gutman {\it et al.} introduced the concepts of energy $\En(G)$ and Laplacian energy $\EnL(G)$ for a simple graph $G$, and furthermore, they proposed a conjecture that for every graph $G$, $\En(G)$ is not more than $\EnL(G)$. Unfortunately,…

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. It is proved that E(G)>= 2(n-\chi(\bar{G}))>= 2(ch(G)-1) for every graph G of order n, and that E(G)>= 2ch(G) for all graphs G…

In 1970s, Gutman introduced the concept of the energy $\En(G)$ for a simple graph $G$, which is defined as the sum of the absolute values of the eigenvalues of $G$. This graph invariant has attracted much attention, and many lower and upper…

Graph energy is the energy of the matrix representation of the graph, where the energy of a matrix is the sum of singular values of the matrix. Depending on the definition of a matrix, one can contemplate graph energy, Randi\'c energy,…

Let $R$ be a Hermitian matrix. The energy of $R$, $\mathcal{E}(R)$, corresponds to the sum of the absolute values of its eigenvalues. In this work it is obtained two lower bounds for $\mathcal{E}(R).$ The first one generalizes a lower bound…

Given a simple graph $G$, its Laplacian-energy-like invariant $LEL(G)$ and incidence energy $IE(G)$ are the sum of square root of its all Laplacian eigenvalues and signless Laplacian eigenvalues, respectively. Applying the Cauchy-Schwarz…

For a given graph \( G \), let \( G^{(j)} \) denote the graph obtained by the deletion of vertex \( v_j \) from \( G \). The difference \( \mathscr{E}(G) - \mathscr{E}(G^{(j)}) \) quantifies the change in the energy of \( G \) upon the…

For a graph $G$, let $S(G)$ be the Seidel matrix of $G$ and $\te_1(G),...,\te_n(G)$ be the eigenvalues of $S(G)$. The Seidel energy of $G$ is defined as $|\te_1(G)|+...+|\te_n(G)|$. Willem Haemers conjectured that the Seidel energy of any…

We derive a two-term asymptotic expansion for the exchange energy of the free electron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent)…

A complex unit gain graph ($ \mathbb{T} $-gain graph), $ \Phi=(G, \varphi) $ is a graph where the gain function $ \varphi $ assigns a unit complex number to each orientation of an edge of $ G $ and its inverse is assigned to the opposite…

The study of spins and particles on graphs has broad applications, from the dynamics of interacting systems on networks to combinatorial problems. Here, we study the large-$n$ limit of the $O(n)$ model on graphs, which is considerably more…

Let $G$ be a simple undirected $n$-vertex graph with the characteristic polynomial of its Laplacian matrix $L(G)$, $\det (\lambda I - L (G))=\sum_{k = 0}^n (-1)^k c_k \lambda^{n - k}$. Laplacian--like energy of a graph is newly proposed…

We show the existence of invariant energy levels in a Kondo lattice model on an isolated complete graph, such as a triangle and a tetrahedron. These energy levels always have fixed eigenenergies $t \pm J/2$, irrespective of the…

Given a graph $M,$ path eigenvalues are eigenvalues of its path matrix. The path energy of a simple graph $M$ is equal to the sum of the absolute values of the path eigenvalues of the graph $M$ (Shikare et. al, 2018). We have discovered new…

Motivated by the linear time algorithm that locates the eigenvalues of a cograph G [10], we investigate the multiplicity of eigenvalue for \lambda \neq -1,0. For cographs with balanced cotrees we determine explicitly the highest value for…

Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randi\'c matrix ${\bf R}=(r_{ij})$ of $G$ is defined by $r_{ij} = 1 / \sqrt{d_jd_j}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and $r_{ij}=0$…

The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by $C_n$ the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of…

Let $G$ be a graph on $n$ vertices with independence number $\alpha(G)$. Let $\mathcal{E}(G)$ be the energy of a graph, defined as the sum of the absolute values of the adjacency eigenvalues of $G$. Using Graffiti, Fajtlowicz conjectured in…